For the simplest of reasons, it doesn't match :)

With re 'debug'

YAPE::Regex::Explaination

One problem I think I see is trying to use look-ahead as look-behind (?!pattern)

I would try something simple, like

my $sequence_to_parse =">test\nATG\nGGG";
while ( $sequence_to_parse =~ m/(^>.+)|(^.+)/gm ) {
    if ( defined $1 ) {
        print "got first line \$1 ($1)\n";
    }
    elsif ( defined $2 ) {
        print "got other line \$2 ($2)\n";
    }
    else {
        print "UH OH \n";
    }
}
__END__
got first line $1 (>test)
got other line $2 (ATG)
got other line $2 (GGG)
[download]

To get just the last line, since lines are chars not \r\n,

my $sequence_to_parse =">test\nATG\nGGG";
print $sequence_to_parse  =~ /([^\r\n]+)$/s;
__END__
GGG
[download]

use YAPE::Regex::Explain;
print YAPE::Regex::Explain->new(
qr/([^\r\n]+)$/s
)->explain;
__END__
The regular expression:

(?s-imx:([^\r\n]+)$)

matches as follows:

NODE                     EXPLANATION
----------------------------------------------------------------------
(?s-imx:                 group, but do not capture (with . matching
                         \n) (case-sensitive) (with ^ and $ matching
                         normally) (matching whitespace and #
                         normally):
----------------------------------------------------------------------
  (                        group and capture to \1:
----------------------------------------------------------------------
    [^\r\n]+                 any character except: '\r' (carriage
                             return), '\n' (newline) (1 or more times
                             (matching the most amount possible))
----------------------------------------------------------------------
  )                        end of \1
----------------------------------------------------------------------
  $                        before an optional \n, and the end of the
                           string
----------------------------------------------------------------------
)                        end of grouping
----------------------------------------------------------------------
[download]

                  my $sequence_to_parse =">test\nATG\nGGG";

while ($sequence_to_parse=~/^>.*\n(^(?!>).*\n.*$)+/gm) {print "$&\n";}
[download]